There are two geometrical structures in a manifold of probability distributions. One is invariant, based on the Fisher information, and the other is based on the Wasserstein distance of optimal transportation. We propose a unified framework which connects the Wasserstein distance and the Kullback-Leibler (KL) divergence to give a new information-geometrical theory. We consider the discrete case consisting of n elements and study the geometry of the probability simplex Sn-1, the set of all probability distributions over n atoms. The Wasserstein distance is introduced in Sn-1 by the optimal transportation of commodities from distribution p ∈ Sn-1 to q ∈ Sn-1. We relax the optimal transportation by using entropy, introduced by Cuturi (2013) and show that the entropy-relaxed transportation plan naturally defines the exponential family and the dually flat structure of information geometry. Although the optimal cost does not define a distance function, we introduce a novel divergence function in Sn-1, which connects the relaxed Wasserstein distance to the KL-divergence by one parameter.
CITATION STYLE
Karakida, R., & Amari, S. I. (2017). Information geometry of wasserstein divergence. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10589 LNCS, pp. 119–126). Springer Verlag. https://doi.org/10.1007/978-3-319-68445-1_14
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